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Magazine Chapter 7: Problem 34
Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$-4 x^{2}+8 x+y^{2}+6 y=-6$$
Short Answer
Expert verified
The equation represents a hyperbola because of differing signs in squared terms.
Step by step solution
01
Rearrange the Equation
Begin by rearranging the equation \[-4x^2 + 8x + y^2 + 6y = -6\] so that similar terms are grouped together. This gives: \[-4(x^2 - 2x) + (y^2 + 6y) = -6.\] Notice we group the \(x\) and \(y\) terms separately.
02
Complete the Square for the x Terms
Complete the square for the \(x\) terms, \(-4(x^2 - 2x)\). The expression \((x^2 - 2x)\) can be rewritten by completing the square: 1. Take half of 2, the coefficient of \(x\) (ignoring the sign), which is 1.2. Square it, getting 1.3. Rewrite: \((x^2 - 2x) = (x - 1)^2 - 1\).Now substitute back: \[-4((x - 1)^2 - 1) = -4(x - 1)^2 + 4.\] The equation becomes:\[-4(x - 1)^2 + y^2 + 6y = -6 + 4.\] Which simplifies to: \[-4(x - 1)^2 + y^2 + 6y = -2.\]
03
Complete the Square for the y Terms
Now complete the square for the \(y\) terms, \(y^2 + 6y\). 1. Take half of 6, the coefficient of \(y\), which is 3.2. Square it, getting 9.3. Rewrite: \(y^2 + 6y = (y + 3)^2 - 9\).Substitute back: \[(y + 3)^2 - 9.\] The equation now reads:\[-4(x - 1)^2 + (y + 3)^2 - 9 = -2.\] Reorder to find:\[-4(x - 1)^2 + (y + 3)^2 = 7.\]
04
Determine the Conic Type
The equation is now \[-4(x - 1)^2 + (y + 3)^2 = 7,\] which shows a negative coefficient for the squared \(x\) term and a positive coefficient for the squared \(y\) term. This indicates a hyperbola, because in the standard form of a hyperbola, \[ rac{(y - k)^2}{a^2} - rac{(x - h)^2}{b^2} = 1\] (replace \(a^2\) and \(b^2\) accordingly) the signs differ.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a crucial algebraic technique used to transform quadratic expressions. It is especially useful when dealing with conic sections such as circles, parabolas, ellipses, and hyperbolas, as it rewrites quadratic equations into a form that is easier to analyze. To complete the square, you follow a few steps:
- Identify the quadratic expression, often located in terms involving x or y, such as in \(x^2 - 2x\) or \(y^2 + 6y\).
- Focus on one variable at a time. Take the coefficient of the linear term (e.g., -2 for x or 6 for y), divide it by 2, and then square it.
- Add and subtract this square inside the expression to maintain equality.
Hyperbola
A hyperbola is a type of conic section characterized by its distinctive open-curved shapes, much like two mirrored parabolas facing away from each other. Unlike circles and ellipses, which have a summation in their standard form, hyperbolas have a subtraction, which reflects their structure involving a transverse axis and a conjugate axis.
- The general form for a hyperbola, with center \( (h, k) \), is \( \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \) or \ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \, depending on its orientation.
- The differentiation of signs between the terms is crucial: one is positive (the leading term) and the other negative, denoting the opening direction of the hyperbola.
Equation Rearrangement
Rearranging an equation is a foundational algebraic skill that helps in transforming an expression into a more useful or readable form, especially for identifying features of conic sections.
- Begin by sorting terms and grouping them according to the variables they contain, as done in \(-4(x^2 - 2x) + (y^2 + 6y) = -6\).
- Completing the square makes these groups clearer, revealing important characteristics such as the center and axes of conic sections.
- Moreover, ensure all constants are suitably positioned on one side to align the equation with typical conic section formats.
